Comparing \(L(s,\chi)\) with its truncated Euler product and generalization (Q985041)

Let \({\mathbb K}/{\mathbb Q}\) be a number field of degree \(d\) and discriminant \(\Delta\). Let \(N\) be the norm of \({\mathbb K}/{\mathbb Q}\). Let \(\Xi\) be a Hecke Grössencharakter. The conductor \(\mathfrak f\) being fixed (of norm \(q\)), there exists an absolute constant \(C>0\) such that no \(L\)-function \(L(s,\Xi)\) has a zero \(\rho\) in the region \[ \text{Re}\,\rho\geq 1-\frac{C}{\log\max(q\Delta,q\Delta|\text{Im}\,s|)} \] except at most one such \(L\)-function. Let \(\Xi\) be a non-exceptional character with conductor \(\mathfrak f\) of norm \(q>1\). The author proves that \[ L(s,\Xi)\asymp\prod_{N{\mathfrak p}\leq q\Delta|s|}(1-\frac{\Xi({\mathfrak p})}{N{\mathfrak p}^s})^{-1} \] when \(1\geq (\text{Re}\, s-1)\log(q\Delta(2+|s|))\geq -\frac{C}{2}\).